Michael Shulman
classifying cosieve

A cosieve is a morphism A→XA\to X in a 2-category that is both ff and a discrete opfibration. Equivalently, it is a subterminal object in Opf(X)Opf(X). It is easy to check that in CatCat, this is equivalent to saying that AA is a full subcategory of XX such that if a∈Aa\in A and f:a→bf:a\to b, then b∈Ab\in A.

A classifying cosieve is a classifying discrete opfibration which is a cosieve—and hence classifies only cosieves, since cosieves are stable under pullback. We write ζ→Ω\zeta\to \Omega for a classifying cosieve. Clearly any such Ω\Omega is posetal.

A cosieve classifier is a classifying cosieve which classifies all cosieves. In this case one can show, just as for the subobject classifier in a topos, that ζ=1\zeta=1. In CatCat, the “walking arrow” 2\mathbf{2} is a cosieve classifier.

is the largest subobject Ω↪S\Omega\hookrightarrow S such that the pullback of EE to Ω\Omega is a cosieve. (Verifying this is a straightforward argument using the Kripke-Joyal semantics?.) It is thus a classifying cosieve, which is canonically associated to E→SE\to S.

In CatCat, the cosieve classifier 2\mathbf{2} arises from SetSet (or any full subcategory of it containing 00 and 11) in this way.

From cosieve classifiers to subobject classifiers

If XX is groupoidal, then every ff into XX is a cosieve. Therefore, maps from a groupoidal XX into a cosieve classifier Ω\Omega classify all subobjects of XX. Since subobjects of XX are the same as subobjects of its coreJ(X)J(X) if that exists, subobjects of XX can be classified by maps J(X)→ΩJ(X) \to \Omega.

Moreover, if a cosieve classifier Ω\Omega itself has a core, then since J(Ω)J(\Omega) is a coreflection of Ω\Omega into gpd(K)gpd(K), it is a subobject classifier in gpd(K)gpd(K) in a suitable (2,1)-categorical sense. Moreover, since Ω\Omega is posetal, its core (if it exists) is discrete. Thus:

Theorem

If KK is a 2-category having a cosieve classifier and enough groupoids, then disc(K)disc(K) has a subobject classifier.

In particular, if KK also has (discrete) exponentials, then disc(K)disc(K) is a topos.

However, disc(K)disc(K) can have both a subobject classifier and a cosieve classifier without the former being a core of the latter. For instance, in the 2-presheaf 2-topos K=[C,Cat]K=[C,Cat], the category disc(K)disc(K) is the 1-topos of 1-sheaves on the homwise-discrete reflection of CC, but there will not in general be a map in either direction relating its subobject classifier to the cosieve classifier.

A subobject classifier can also be constructed from a cosieve classifier in a Heyting 2-category with a duality involution. For then if Ω\Omega is a cosieve classifier, Ωo\Omega^o is a sieve opclassifier, i.e. K(X,Ωo)K(X,\Omega^o) is equivalent to the opposite of the poset of sieves on XX. On Ω×Ωo\Omega\times\Omega^o we thus have both a sieve RR and a cosieve SS, pulled back from Ω\Omega and Ωo\Omega^o; let Ωd\Omega_d be the subobject of Ω×Ωo\Omega\times\Omega^o defined as R⇔SR\Leftrightarrow S in the Heyting algebra structure. Now maps into Ωd\Omega_d classify sieves and cosieves that are equal as subobjects, which is to say, subobjects that are both sieves and cosieves. And transformations between maps X→ΩdX\to \Omega_d correspond to both inclusions of cosieves and coinclusions of sieves, which is to say, identities; thus Ωd\Omega_d is discrete, and hence a subobject classifier in disc(K)disc(K).

Theorem

If KK is a Heyting 2-category having a cosieve classifier and a duality involution, then disc(K)disc(K) has a subobject classifier.

Replacing subobject classifiers

If KK has a cosieve classifier and (discrete) exponentials, but not enough groupoids, then disc(K)disc(K) may not be a topos. But it retains many of the properties of a topos, because even though the “power object” PX=ΩXP X = \Omega^X is not an object of disc(K)disc(K), it can still be quantified over in the internal logic of KK to define objects and properties in disc(K)disc(K), and even in gpd(K)gpd(K), where all subobjects are cosieves.

For example, if KK is also Heyting, then for any groupoidal XX we can construct the “internally least” subobject of XX with some property, as

This allows the construction of all sorts of “closure” operations that exist in a topos, such as the equivalence relation generated by any given relation on a groupoidal object. In particular:

Proposition

If KK is 1-exact and Heyting with exponentials and a cosieve classifier, then disc(K)disc(K) is finitely cocomplete.

If XX is not groupoidal, then the above technique only constructs cosieves in XX rather than arbitrary subobjects of it. However, if there are enough groupoids, we can construct arbitrary subobjects of any object XX in this way, since subobjects of XX are bijective with subobjects of its core J(X)J(X). In particular:

Proposition

Proof

It suffices to be able to construct the equivalence relation generated by the image of A2→A×AA ^{\mathbf{2}} \to A\times A, for any AA. Note that these relations are not cosieves on A×AA\times A, but as remarked above, we can get around this if A×AA\times A has a core. Alternately, since the relations we care about are all “2-sided sieves” (subterminals in Fib(A,A)Fib(A,A)), if there is a duality involution we can turn them into cosieves on Aop×AA^{op}\times A and perform the closure there.

In another vein, if KK is a positiveHeyting 2-category with a (necessarily discrete) natural numbers objectNN, we can of course construct the discrete object of rational numbers QQ in the usual way, and then define the Dedekind real numbers as two-sided cuts. Thus, RR is a subobject of PQ×PQP Q \times P Q, and hence posetal, but since the order relation on RR inherited from the two copies of PQP Q would go in different directions, in fact RR is discrete.

I haven’t made a concerted effort yet, but I haven’t yet thought of any really important aspect of topos-ness for disc(K)disc(K) that isn’t almost as well-served by having a posetal power-object rather than a discrete one. Mathieu Dupont was the one who originally pointed out to me that 2-categorically, power-objects are naturally posets rather than sets.